Optimal resource allocation in a multi-hop OFDMA wireless network with cooperative relaying

ABSTRACT

An optimal resource allocation strategy for OFDMA multi-hop wireless networks is disclosed. The system allocates one or more resources in a multi-hop network by solving one or more higher-layer sub-problem; solving one or more physical layer and media access control (PHY/MAC) layer sub-problems per tone per time slot with one of cooperative relaying of radio signals or spatial reusing of radio spectrum; updating prices; and allocating radio resources based on the PHY/MAC layer sub-problems.

The present application claims priority to Provisional Application Ser.No. 60/894,550 filed Mar. 13, 2007, the content of which is incorporatedby reference.

BACKGROUND

The present invention relates to optimal resource allocation in amulti-hop OFDMA wireless network with cooperative relaying.

Unlike the wired networks, wireless networks face additional challengesdue to the shared nature of the wireless channel. That is, thecapacities of different links in the wireless networks may be coupled asthe links experience interference from each other. Thus, an optimalresource allocation strategy is required for efficient use of thenetwork resources

Permitting simple receiver design in the frequency-selective channels,OFDMA is a major access technology for the future broadband networks.However, from the viewpoint of radio resource allocation, OFDMA yields amuch more complex problem due to the existence of a large number ofparallel subchannels. The difficulty is aggravated in the interferenceenvironment where multiple links can potentially share the subchannels.In OFDMA systems, the channels are subdivided in the frequency domaininto subchannels, to which the given power resources must be allocated.If more than one links are to be activated in the same subchannelsimultaneously, mutual interference between the links must also beconsidered. Also, if the cooperative relaying technique is to beemployed for enhanced spectral efficiency, the power split between thecooperating relays must be determined while still accounting for theaforementioned interference issue.

In one approach, a closed-form solution was obtained for the power andspectrum allocation problem for multi-hop MIMO cooperative relaynetworks by approximating log(1+x) by √{square root over (x)}. It wasthen verified that the approximate solution is actually quite close tothe exact optimal solution within a wide range of operating SNR.However, the power constraint was imposed on the network-wide totalpower instead of the individual power constraints per node. Also, thework assumed a flat fading channel, and hence may not be suitable forfrequency selective channels that appear in wideband signaling. Inanother approach, the optimal power and rate control problem was solvedfor an OFDMA cooperative cellular network, where the mobiles can act ascooperative relays for other mobile terminals. Depending on the channelconditions perceived by the mobiles on each tone, the mobiles may opt toemploy the cooperative relaying on that tone. However, since thedecision is made on a tone-by-tone basis, the mobiles must be able totransmit on a particular tone at the same time as they listen on anothertone.

SUMMARY

An optimal resource allocation strategy for OFDMA multi-hop wirelessnetworks is disclosed. The system allocates one or more resources in amulti-hop network by solving one or more higher-layer sub-problem;solving one or more physical layer and media access control (PHY/MAC)layer sub-problems per tone per time slot with one of cooperativerelaying of radio signals or spatial reusing of radio spectrum; updatingprices; and allocating radio resources based on the PHY/MAC layersub-problems.

Implementations of the above aspect can include one or more of thefollowing. The process first solves high layer sub problems such as

${\max\limits_{s,s_{\min},x}s_{\min}} - {\lambda^{T}x}$subject  to  s_(min)s Ax = s x ≽ 0.

The system can also solve PHY/MAC sub-problems per tone per time slotwithout cooperative relaying and no spatial reuse such as:

$P_{\ell}^{{({n,t})}^{*}} = \left\lbrack {\frac{\lambda_{\ell}}{\mu_{S{(\ell)}}^{(t)} + \varepsilon} - \frac{1}{G_{\ell\ell}^{(n)}}} \right\rbrack^{+}$$\ell^{*} = {\arg\;{\max\limits_{\ell}\left\{ \left. {{\lambda_{\ell}{\log\left( {1 + {G_{\ell\ell}^{(n)}P_{\ell}^{{({n,t})}^{*}}}} \right)}} - {\mu_{S{(\ell)}}^{(t)}P_{\ell}^{{({n,t})}^{*}}}} \right|_{\ell \in {{??}_{t}{({??})}}} \right\}}}$

The system also solves the PHY/MAC sub-problems per tone per time slot,but with cooperative relaying and no spatial reuse

$P_{\ell}^{{({n,t})}^{*}} = \left\lbrack {\frac{\lambda_{\ell}}{\mu_{S{(\ell)}}^{(t)} + \varepsilon} - \frac{1}{G_{k^{*},{S{(\ell)}}}^{(n)}}} \right\rbrack^{+}$${{\hat{P}}_{\ell}^{{({n,t})}^{*}}\left( k_{i} \right)} = {\frac{1}{\eta_{i}^{2}}\left\lbrack {\frac{\lambda_{\ell}\eta_{i}}{\mu_{k_{i}}^{(t)} + \varepsilon} - \frac{1}{G_{{Q{(\ell)}},k_{i}}^{(n)}}} \right\rbrack}^{+}$$\ell^{*} = {\arg\;{\max_{\ell}{\left\{ {\left. {{\lambda_{\ell}{\log\left( {1 + {G_{\ell\ell}^{(n)}P_{\ell}^{{({n,t})}^{*}}}} \right)}} - {\mu_{S{(\ell)}}^{(t)}P_{\ell}^{{({n,t})}^{*}}}} \right|_{\ell \in {{??}_{t}{({??})}}},\left. {{\lambda_{\ell}{\log\left( {1 + {G_{k^{*},{S{(\ell)}}}^{(n)}P_{\ell}^{{({n,t})}^{*}}}} \right)}} - {\mu_{S{(\ell)}}^{(t)}P_{\ell}^{{({n,t})}^{*}}}} \right|_{\ell \in {{A_{t}{({??}^{\prime})}}\bigcap\mathcal{E}_{b}}},\left. {{\lambda_{\ell}{\log\left( {1 + \left( {\sum\limits_{k \in {\mathcal{R}{(\ell)}}}^{\;}\;\sqrt{G_{{Q{(\ell)}},k}^{(n)}{{\hat{P}}_{\ell}^{{({n,t})}^{*}}(k)}}} \right)^{2}} \right)}} - {\sum\limits_{k \in {\mathcal{R}{(\ell)}}}^{\;}\;{\mu_{k}^{(t)}{{\hat{P}}_{\ell}^{{({n,t})}^{*}}(k)}}}} \right|_{\ell \in {{{??}_{t}{({??}^{\prime})}}\bigcap\mathcal{E}_{c}}}} \right\}.}}}$

Additionally, the system solves the PHY/MAC sub-problems per tone pertime slot, but with no cooperative relaying and with spatial reuse

${{Solve}\mspace{14mu}\max{\sum\limits_{\ell \in {??}_{t}}^{\;}\;{\lambda_{\ell}c_{\ell}^{({n,t})}}}} - {\sum\limits_{k}\;{\left( {\mu_{\ell}^{(t)} + \varepsilon} \right){\sum\limits_{\underset{\ell \in {??}_{t}}{\ell \in {{??}{(k)}}}}\; P_{\ell}^{({n,t})}}}}$${{{subject}\mspace{14mu}{to}\mspace{14mu} P_{\ell}^{({n,t})}} \geq 0},{c_{\ell}^{({n,t})} \leq {\log\left( {1 + \frac{G_{\ell\ell}^{(n)}P_{\ell}^{({n,t})}}{{\sum\limits_{\substack{\ell^{\prime} \neq \ell \\ \ell^{\prime} \in {??}_{t}}}^{\;}\;{G_{{\ell\ell}^{\prime}}^{(n)}P_{\ell^{\prime}}^{({n,t})}}} + 1}} \right)}}$

Additionally, the process solves the PHY/MAC sub-problems per tone pertime slot, but with cooperative relaying and spatial reuse

${{{\max{\sum\limits_{\ell \in {{??}_{t}{({??}^{\prime})}}}^{\;}\;{\lambda_{\ell}c_{\ell}^{({n,t})}}}} - {\sum\limits_{k}^{\;}\;{\left( {\mu_{\ell}^{(t)} + \varepsilon} \right)\begin{pmatrix}{{\sum\limits_{\substack{\ell \in {{??}{(k)}} \\ \ell \in {{({\mathcal{E}\bigcup\mathcal{E}_{b}})}\bigcap{{??}_{t}{({??}^{\prime})}}}}}^{\;}\; P_{\ell}^{({n,t})}} +} \\{\sum\limits_{\substack{\ell \in {\mathcal{E}_{c}\bigcap{{??}_{t}{({??}^{\prime})}}} \\ k \in {\mathcal{R}{(\ell)}}}}^{\;}\;{{\hat{P}}_{\ell}^{({n,t})}(k)}}\end{pmatrix}{s.t.c_{\ell}^{({n,t})}}}}} \leq {\min\limits_{k \in {\mathcal{R}{(\ell)}}}c_{\ell,k}^{({n,t})}}},{\forall{\ell \in {{{??}_{t}\left( {??}^{\prime} \right)}\bigcap\mathcal{E}_{b}}}},{c_{\ell,k}^{({n,t})}\overset{\Delta}{=}{{\log\left( {1 + \frac{G_{k,{S{(\ell)}}}^{(n)}P_{\ell}^{({n,t})}}{\begin{matrix}{{\sum\limits_{\substack{\ell^{\prime} \neq \ell \\ \ell^{\prime} \in {{({\mathcal{E}\bigcup\mathcal{E}_{b}})}\bigcap{{??}_{t}{({??}^{\prime})}}}}}^{\;}\;{G_{k,{S{(\ell^{\prime})}}}^{(n)}P_{\ell^{\prime}}^{({n,t})}}} +} \\{{\sum\limits_{\substack{\ell^{\prime} \neq \ell \\ \ell^{\prime} \in {\mathcal{E}_{c}\bigcap{{??}_{t}{({??}^{\prime})}}}}}^{\;}\;{\sum\limits_{k^{\prime} \in {\mathcal{R}{(\ell^{\prime})}}}^{\;}\;{G_{k,k^{\prime}}^{(n)}{{\hat{P}}_{\ell^{\prime}}^{({n,t})}\left( k^{\prime} \right)}}}} + 1}\end{matrix}}} \right)}{c_{\ell}^{({n,t})} \leq {\log\left( {1 + \frac{\left( {\sum\limits_{k \in {\mathcal{R}{(\ell)}}}^{\;}\;\sqrt{G_{{Q{(\ell)}},k}^{(n)}{{\hat{P}}_{\ell}^{({n,t})}(k)}}} \right)^{2}}{\begin{matrix}{{\sum\limits_{\substack{\ell^{\prime} \neq \ell \\ \ell^{\prime} \in {{({\mathcal{E}\bigcup\mathcal{E}_{b}})}\bigcap}}}^{\;}\;{G_{{Q{(\ell)}},{S{(\ell^{\prime})}}}^{(n)}P_{\ell^{\prime}}^{({n,t})}}} +} \\{{\sum\limits_{\substack{\ell^{\prime} \neq \ell \\ \ell^{\prime} \in {\mathcal{E}_{c}\bigcap{{??}_{t}{({??}^{\prime})}}}}}^{\;}\;{\sum\limits_{k^{\prime} \in {\mathcal{R}{(\ell^{\prime})}}}^{\;}\;{G_{{Q{(\ell)}},k^{\prime}}^{(n)}{{\hat{P}}_{\ell^{\prime}}^{({n,t})}\left( k^{\prime} \right)}}}} + 1}\end{matrix}}} \right)}}}},{\forall{\ell \in {{{{??}_{t}\left( {??}^{\prime} \right)}\bigcap{\mathcal{E}_{c}c_{\ell}^{({n,t})}}} \leq {\log\left( {1 + \frac{G_{\ell\ell}^{(n)}P_{\ell}^{({n,t})}}{\begin{matrix}{{\sum\limits_{\substack{\ell^{\prime} \neq \ell \\ \ell^{\prime} \in {{({\mathcal{E}\bigcup\mathcal{E}_{b}})}\bigcap{{??}_{t}{({??}^{\prime})}}}}}^{\;}\;{G_{{\ell\ell}^{\prime}}^{(n)}P_{\ell^{\prime}}^{({n,t})}}} +} \\{{\sum\limits_{\substack{\ell^{\prime} \neq \ell \\ \ell^{\prime} \in {\mathcal{E}_{c}\bigcap{{??}_{t}{({??}^{\prime})}}}}}^{\;}\;{\sum\limits_{k^{\prime} \in {\mathcal{R}{(\ell^{\prime})}}}^{\;}\;{G_{{Q{(\ell)}},k^{\prime}}^{(n)}{{\hat{P}}_{\ell^{\prime}}^{({n,t})}\left( k^{\prime} \right)}}}} + 1}\end{matrix}}} \right)}}}},{\forall{\ell \in {{??}_{t}({??})}}}$

Prices are updated, and the process checks for convergence untilconvergence is achieved. The overall cross-layer optimization problemcan be first decomposed into different layers using the dualdecomposition technique. The system solves the PHY/MAC subproblem as anon-convex optimization problem. Modeling with and without cooperativerelaying as well as per-tone decomposition can be performed. Given theformulations, the methods to solve the per-tone PHY/MAC subproblemsincluding various practical issues regarding the optimizationalgorithms. In particular, closed-form expressions can be obtained forthe optimal solutions in some special cases.

Advantages of the preferred embodiments may include one or more of thefollowing. The preferred embodiment provides a centralized resourceallocation method that can yield optimal or close-to-optimal end-to-endthroughput in such a setting. Optimal power allocation in the networkwith pure OFDMA (that is, without spatial reuse of OFDM tones) itself isan advanced technique that can readily boost the throughputsignificantly. The system provides a closed-form solution for thePHY/MAC subproblem in this case, which makes the overall algorithmfaster. The system allows spatial reuse and cooperative relayingenhancements. The cross-layer optimization problem is formulated thatmaximizes the balanced end-to-end throughput under the routing and thePHY/MAC constraints. A dual method is employed to solve the problemefficiently and optimally by decoupling it into different layers as wellas into different OFDM tones. A cooperative relaying technique isincorporated into the framework to improve the performance. Mutualinterference between the links is explicitly modeled to allow maximalspatial reuse of the spectral resources, and half-duplex operation ofthe radios is assumed. The numerical results exhibit how the bottleneckphenomenon typical in multi-hop networks can be alleviated by theproposed techniques to yield significant improvement in the throughput.The preferred embodiment provides an efficient method to solve theresource allocation problem for multi-hop wireless networks that useOFDMA technology. The preferred embodiment solves the optimal resourceallocation problem in multi-hop wireless networks with an OFDMA airinterface that can exploit spatial reuse and cooperative relaying. Thesystem incorporates the cooperative relaying technique into thecross-layer design. In multi-hop networks, natural avenues can be usedfor exploiting cooperative communication techniques due to the relayingoperation involved in routing the packets in the networks. The multi-hoprouting constraints are explicitly modeled. The mutual co-channelinterference is properly accounted for even with cooperative relayingwhile strict half-duplex constraints are enforced. The system provides asimple scheduling framework that preserves strict half-duplex operation.The system allows for a spatial reuse of the spectrum. That is, when aparticular set of links impose relatively low interference to each otheron a particular band, it is possible to activate those linkssimultaneously to maximize the spectral efficiency of the network. Thissituation occurs more frequently in the multi-hop networks as thenetwork size grows, and is efficiently exploited.

BRIEF DESCRIPTION OF THE DRAWINGS

FIGS. 1-2 show diagrams illustrating a wireless communication networksuitable for implementing one embodiment of the present invention.

FIG. 3 shows one exemplary process for performing optimal resourceallocation in a multi-hop OFDMA wireless network with cooperativerelaying.

FIG. 4 shows an exemplary cooperative relaying scheme.

DESCRIPTION

FIGS. 1-2 show diagrams illustrating a wireless OFDMA communicationnetwork suitable for implementing one embodiment of the presentinvention. In OFDMA systems, the channels are subdivided in thefrequency domain into subchannels, to which the given power resourcesmust be allocated. If more than one links are to be activated in thesame subchannel simultaneously, mutual interference between the linksmust also be considered. Also, if the cooperative relaying technique isto be employed for enhanced spectral efficiency, the power split betweenthe cooperating relays must be determined while still accounting for theaforementioned interference issue. Then, the goal of the resourceallocation is to enhance the end-to-end throughput which measures thetraffic rate from each of the nodes in the network that can be deliveredto the gateway. To achieve this, how to route the traffic from the nodesin a multi-hop fashion must be jointly considered.

Turning now to FIG. 1, the wireless communication network includes aradio resource controller 10 receiving channel measurement data from aplurality of base stations BS. The base stations communicate with eachother through one or more wireless links. The base stations also receiveradio resource allocation commands from the radio resource controller10.

Referring now to FIG. 2, the wireless network has, illustratively, aplurality of base stations BS1 . . . BSB, each having one or morewireless antennas. Each base station BS includes a media access control(MAC) layer and a physical layer (PHY) layer. The base station BS inturn communicates with one or more mobile stations MS1 . . . MSK, eachof which can communicate with different base stations at differenttimes.

Base stations BS illustratively have an array of multiple cooperatingantennas so as to take advantage of multiple input, multiple output(MIMO) techniques, as discussed above. Each BS node is, for example andwithout limitation, a node in a wireless multi-hop network. In order tocommunicate with one another, the base stations BS of FIG. 1 may use themultiple antennas in those nodes to perform adaptive beamforming. Aswill be familiar to one skilled in the art, adaptive beamforming is atechnique in which an array of antennas is able to achieve maximumreception in a specific angular direction by estimating the signalarrival from a desired direction (in the presence of noise) whilesignals of the same frequency from other directions are rejected. Thisis achieved by varying the weights of each of the transmitters(antennas) used in the array. The spatial separation of arriving signalsis exploited to separate the desired signal from the interferingsignals. In adaptive beamforming the optimum weights of the antennas aretypically iteratively computed. As is also known to one skilled in theart, beamforming is generally accomplished by phasing the feed to eachelement of an array so that signals received or transmitted from allelements will be in phase in a particular direction.

FIG. 3 shows one exemplary process for performing optimal resourceallocation in a multi-hop OFDMA wireless network with cooperativerelaying. In this process, the following annotations are used:

x: traffic flow vector

s: vector of end-to-end throughput

P_(l) ^((n,t)): transmit power for link l on tone n in time slot t

{circumflex over (P)}_(l) ^((n,t))(k): transmit power at cooperativerelay k associated with (virtual) link l on tone n in time slot t

c_(l) ^((n,t)): link capacity of link l on tone n in time slot t

P_(k,max) maximum transmit power at node k

λ: Lagrange multiplier (or relative weight) for link l

μ: Lagrange multiplier associated with power constraints

The process first solves high layer sub problems (50).

${\max\limits_{x,s,s_{\min}}s_{\min}} - {\lambda^{T}x}$

subject tos_(min)sAx=sx

0

Next, the process solves PHY/MAC sub-problems per tone per time slot(52) if cooperative relaying with no spatial reuse is employed:

$P_{l}^{{({n,t})}^{*}} = \left\lbrack {\frac{\lambda_{l}}{\mu_{S{(l)}}^{(t)} + \varepsilon} - \frac{1}{G_{ll}^{(n)}}} \right\rbrack^{+}$$l^{*} = {\arg\;{\max\limits_{l}\left\{ {{{\lambda_{l}{\log\left( {1 + {G_{ll}^{(n)}P_{l}^{{({n,t})}^{*}}}} \right)}} - {\mu_{S{(l)}}^{(t)}P_{l}^{{({n,t})}^{*}}}}❘_{l \in}} \right\}}}$

Alternatively, the process solves the PHY/MAC sub-problems per tone pertime slot, but with cooperative relaying and no spatial reuse (54):

$P_{l}^{{({n,t})}^{*}} = \left\lbrack {\frac{\lambda_{l}}{\mu_{S{(l)}}^{(t)} + \varepsilon} - \frac{1}{G_{k^{*},{S{(l)}}}^{(n)}}} \right\rbrack^{+}$${{\hat{P}}_{l}^{{({n,t})}^{*}}\left( k_{i} \right)} = {\frac{1}{\eta_{i}^{2}}\left\lbrack {\frac{\lambda_{l}\eta_{i}}{\mu_{k_{i}}^{(t)} + \varepsilon} - \frac{1}{G_{{Q{(l)}},k_{i}}^{(n)}}} \right\rbrack}^{+}$$l^{*} = {\arg\;{\max_{l}{\left\{ {{{{\lambda_{l}{\log\left( {1 + {G_{ll}^{(n)}P_{l}^{{({n,t})}^{*}}}} \right)}} - {\mu_{S{(l)}}^{(t)}P_{l}^{{({n,t})}^{*}}}}❘_{l \in {{??}_{??}{({??})}}}},\left. {{\lambda_{l}{\log\left( {1 + {G_{k^{*},{S{(l)}}}^{(n)}P_{l}^{{({n,t})}^{*}}}} \right)}} - {\mu_{S{(l)}}^{(t)}P_{l}^{{({n,t})}^{*}}}} \right|_{l \in {{{??}_{??}{({??}^{\prime})}}\bigcap\; ɛ_{b}}},\left. {{\lambda_{l}{\log\left( {1 + \left( {\sum\limits_{k \in {\mathcal{R}{(l)}}}\sqrt{G_{{Q{(l)}},k}^{(n)}{{\hat{P}}_{l}^{{({n,t})}^{*}}(k)}}} \right)^{2}} \right)}} - {\sum\limits_{k \in {\mathcal{R}{(l)}}}{\mu_{k}^{(t)}{{\hat{P}}_{l}^{{({n,t})}^{*}}(k)}}}} \right|_{l \in {{{??}_{??}{({??}^{\prime})}}\bigcap\; ɛ_{c}}}} \right\}.}}}$

Alternatively, the process solves the PHY/MAC sub-problems per tone pertime slot, but with no cooperative relaying and with spatial reuse (56):

${{Solve}\mspace{14mu}\max{\sum\limits_{l \in {??}_{??}}{\lambda_{l}c_{l}^{({n,t})}}}} - {\sum\limits_{k}{\left( {\mu_{l}^{(t)} + \varepsilon} \right){\sum\limits_{\underset{l \in {??}_{??}}{l \in {{??}{(k)}}}}P_{l}^{({n,t})}}}}$${{{subject}\mspace{14mu}{to}\mspace{14mu} P_{l}^{({n,t})}} \geq 0},{c_{l}^{({n,t})} \leq {\log\left( {1 + \frac{G_{ll}^{(n)}P_{l}^{({n,t})}}{{\sum\limits_{\underset{l^{\prime} \in {??}_{??}}{l^{\prime} \neq l}}{G_{{ll}^{\prime}}^{(n)}P_{l^{\prime}}^{({n,t})}}} + 1}} \right)}}$

Alternatively, the process solves the PHY/MAC sub-problems per tone pertime slot, but with cooperative relaying and spatial reuse (58):

${{{\max{\sum\limits_{l \in {{??}_{??}{({??}^{\prime})}}}{\lambda_{l}c_{l}^{({n,t})}}}} - {\sum\limits_{k}{\left( {\mu_{l}^{(t)} + \varepsilon} \right)\left( {{\sum\limits_{\underset{l \in {{({ɛ\bigcup\; ɛ_{b}})}\bigcap{{??}_{??}{({??}^{\prime})}}}}{l \in {{??}{(k)}}}}P_{l}^{({n,t})}} + {\sum\limits_{\underset{k \in {\mathcal{R}{(l)}}}{l \in \;{ɛ_{c}\bigcap{{??}_{??}{({??}^{\prime})}}}}}{{\hat{P}}_{l}^{({n,t})}(k)}}} \right){s.t.c_{l}^{({n,t})}}}}} \leq {\min\limits_{k \in \;{\mathcal{R}{(l)}}}c_{l,k}^{({n,t})}}},{\forall{l \in {{{??}_{??}\left( {??}^{\prime} \right)}\bigcap\; ɛ_{b}}}},{c_{l,k}^{({n,t})}\overset{\Delta}{=}{\log\left( {1 + \frac{G_{k,{S{(l)}}}^{(n)}P_{l}^{({n,t})}}{\begin{matrix}{{\sum\limits_{\underset{l^{\prime} \in {{({ɛ\bigcup\; ɛ_{b}})}\bigcap{{??}_{??}{({??}^{\prime})}}}}{l^{\prime} \neq l}}^{\;}{G_{k,{S{(l^{\prime})}}}^{(n)}P_{l^{\prime}}^{({n,t})}}} +} \\{{\sum\limits_{\substack{l^{\prime} \neq l \\ l^{\prime} \in \;{ɛ_{c}\bigcap{{??}_{??}{({??}^{\prime})}}}}}{\sum\limits_{k^{\prime} \in {\mathcal{R}{(l^{\prime})}}}{G_{k,k^{\prime}}^{(n)}{{\hat{P}}_{l^{\prime}}^{({n,t})}\left( k^{\prime} \right)}}}} + 1}\end{matrix}}} \right)}}$${c_{l}^{({n,t})} \leq {\log\left( {1 + \frac{\left( {\sum\limits_{k \in {\mathcal{R}{(l)}}}\sqrt{G_{{Q{(l)}},k}^{(n)}{{\hat{P}}_{l}^{({n,t})}(k)}}} \right)^{2}}{\begin{matrix}{{\sum\limits_{\underset{l^{\prime} \in {{({ɛ\;\bigcup\; ɛ_{b}})}\bigcap{{??}_{??}{({??}^{\prime})}}}}{l^{\prime} \neq l}}{G_{{Q{(l)}},{S{(l^{\prime})}}}^{(n)}P_{l^{\prime}}^{({n,t})}}} +} \\{{\sum\limits_{\underset{l^{\prime} \in \;{ɛ_{c}\bigcap{{??}_{??}{({??}^{\prime})}}}}{l^{\prime} \neq l}}{\sum\limits_{k^{\prime} \in {\mathcal{R}{(l^{\prime})}}}{G_{{Q{(l)}},k^{\prime}}^{(n)}{{\hat{P}}_{l^{\prime}}^{({n,t})}\left( k^{\prime} \right)}}}} + 1}\end{matrix}}} \right)}},{\forall{l \in {{{??}_{??}\left( {??}^{\prime} \right)}\bigcap ɛ_{c}}}}$${c_{l}^{({n,t})} \leq {\log\left( {1 + \frac{G_{ll}^{(n)}P_{l}^{({n,t})}}{\begin{matrix}{{\sum\limits_{\underset{l^{\prime} \in {{({ɛ\bigcup\; ɛ_{b}})}\bigcap{{??}_{??}{({??}^{\prime})}}}}{l^{\prime} \neq l}}{G_{{ll}^{\prime}}^{(n)}P_{l^{\prime}}^{({n,t})}}} +} \\{{\sum\limits_{\underset{l^{\prime} \in \;{ɛ_{c}\bigcap{{??}_{??}{({??}^{\prime})}}}}{l^{\prime} \neq l}}{\sum\limits_{k^{\prime} \in \;{\mathcal{R}{(l^{\prime})}}}{G_{{Q{(l)}},k^{\prime}}^{(n)}{{\hat{P}}_{l^{\prime}}^{({n,t})}\left( k^{\prime} \right)}}}} + 1}\end{matrix}}} \right)}},{\forall{l \in {{??}_{??}({??})}}}$

Prices are updated (60), and the process checks for convergence (62) anduntil convergence is achieved it loops back to 50. The system of FIGS.3A-3B takes into account the cross-layer interaction in an optimalfashion and solves the PHY/MAC problem with spatial reuse and/orcooperative relaying.

In the multi-hop network consisting of K base stations (BSs) and agateway node with the radio resource controller 10, where the gatewaynode is connected to the core network via a high-capacity wireline link.The BSs function as the access points that aggregate the user data to betransported to the gateway node. The BSs form a mesh topology to achievemulti-hop connectivity to the gateway by means of a network entry andtopology formation protocol. It is assumed that this topology hasalready been formed and fixed. The network topology can be modeled as adirected graph

=(

,

), where

is the set of K+1 nodes representing the BSs and the gateway node, and

is the set of L directed edges representing the links between the BSs.For each edge lε

, the head (transmitter) is denoted by S(l), and the tail (receiver) isdenoted by

(l). Also, denote the set of edges that start from a node kε

by

(k), and the set of edges entering a node k by

(k). To simplify the exposition, it is assumed that the traffic is onlyin the upstream direction from the BSs to the gateway and no traffic isinjected from the gateway toward the BSs.

The radios installed in the nodes are assumed to operate in ahalf-duplex mode. That is, a radio cannot transmit and receive at thesame time. For practical concerns, the duplexing mode is assumed to bethe same for all the OFDM subchannels within a radio. Thus, when a radiois in the transmission [reception] mode, it cannot receive [transmit] inany of its subchannels.

Due to the half-duplex assumption, an appropriate link scheduling schememust be introduced to formulate the problem. Here, the “even-odd”scheduling framework is used to address a QoS-guaranteed multi-hopnetwork optimization problem.

Consider a frame consisting of T time slots. To simplify the exposition,it is assumed that there are only two time slots within a frame (T=2)and each slot is indexed by the slot number t=0 or 1. The idea of the“even-odd” framework is to partition the nodes in the network into twogroups such that by scheduling those groups in different time slots, thehalf-duplex constraint is satisfied. In a tree-topology network, thiscan be easily accomplished by grouping the nodes based on their hopdistances from the gateway: if the hop distance of a node is an even[odd] number, the node belongs to the “even” [“odd”] group. Then, allthe nodes in the “even” group are activated (i.e., transmit) in timeslot 0, when all the nodes in the “odd” group listen. In the next timeslot t=1, the nodes in the “odd” group are activated.

This framework can be extended to the mesh topology if a certainconstraint is imposed on the topology. The difference of the meshtopology from the tree topology is that there can be more than one routebetween each node and the gateway. In order to make the grouping rulebased on the hop count unambiguous, the hop counts along any of thosemultiple routes must have the same remainder when divided by 2. Thenetwork topology formation protocol ensures that this is the case.Therefore, even in a mesh-topology network, the half-duplex constraintcan be met by activating the “even” group of nodes in time slot t=0 andthe “odd” group of nodes in time slot t=1.

Note that activating a node k implies that all the links emanating fromthat node

(k) are activated. Denote the set of links activated in time slot t by

_(t). The links activated in the same time slot may be multiplexed overdifferent OFDM tones to avoid interference totally, which corresponds tothe pure OFDMA without spatial reuse. In a cross-layer setting, theoptimization space is quite large even for this simple strategy, as foreach tone and time slot, a link must be chosen with appropriate transmitpower and rate to satisfy the routing and the power constraints andoptimize a higher-layer performance objective. A centralized networkcontroller that first collects the frequency-selective channel gains andruns the optimization algorithm to obtain the optimal solutions.

When spatial resource reuse is to be exploited by activating more thanone links in an OFDM tone, one has to deal with the co-channelinterference, which often makes the relevant physical layer optimizationproblem non-convex. Also, when cooperative relaying strategy is to beemployed, one has to find out which links must cooperative over whichtones in each time slot. Apparently, it is a challenging optimizationproblem, where it is important to keep the complexity of theoptimization algorithm contained.

The cross-layer optimization problems are often formulated as networkutility maximization (NUM) problems, where the aim is to maximize aconcave utility function of the end-to-end flows under the specificconstraints arising from the limited network resources. Here, thecross-layer optimization problem is formulated to maximize the minimumthroughput among all BSs under the routing and the PHY constraints. Thisensures a fair throughput is achieved from each BS to the gateway,regardless of the actual hop distances to the gateway. The fairness isan important performance goal in multi-hop wireless networks. The systemthen solves an optimization problem given by

-   -   subject to        max s_(min)  (1)        s_(min)≦s  (2)        xε        (s)  (3)        x≦c  (4)        cε          (5)

where the k-th element s_(k) of s represents the throughput of the flowfrom BS k to the gateway, s_(min) the minimum of s_(k), k=1, 2, . . . ,K as is enforced by the constraint, the l-th element x_(l) of x is thedata flow rate over link l, and the l-th element c_(l) of c is the PHYcapacity of link l. The network flow region

is characterized by the network topology and the routing constraints.Eq. 3 requires that the flow rate vector x should obey theseconstraints. The PHY capacity region

is governed by the PHY/MAC resource constraints as well as the specificPHY/MAC schemes employed to establish the wireless connections. Eq. 5enforces that the link capacities lie within the supportable PHY/MACrate region.

It can be seen that the only coupling constraint between the PHY/MAClayer and the higher layers is (4), which states that the link flow ratecannot exceed the capacity provided by the PHY/MAC layer. A layeredapproach is possible by solving the problem in the dual domainintroducing a Lagrange multiplier vector λ to relax this constraint. Thepartial Lagrangian can be written as

_(layer)(s,s _(min) ,x,c,λ)=s _(min)−λ^(T)(x−c).  (1)

The dual objective function is then given by

$\begin{matrix}\begin{matrix}{{{g(\lambda)} = {\max\limits_{s,s_{\min},x,c}{{\mathcal{L}_{layer}\left( {s,s_{\min},x,c,\lambda} \right)}\mspace{11mu}{subject}\mspace{14mu}{to}\mspace{14mu}(2)}}},(3),{{and}\mspace{11mu}(5)}} \\{= {{\max\limits_{s,s_{\min},x}\left\{ {\left. {s_{\min} - {\lambda^{T}x}} \middle| {s_{\min} \preccurlyeq s} \right.,{x \in {\mathcal{F}(s)}}} \right\}} +}} \\{\max\limits_{c}\left\{ {\lambda^{T}c} \middle| {c \in {??}} \right\}}\end{matrix} & \begin{matrix}(7) \\(8)\end{matrix}\end{matrix}$

In (8), it is evident that the problem is now decoupled into the higherlayer (the first term) and the PHY/MAC layer (the second term). Theoptimal Lagrange multiplier is obtained by solving the dual problem

$\begin{matrix}{\min\limits_{\lambda \succcurlyeq 0}{{g(\lambda)}.}} & (9)\end{matrix}$

Provided that the duality gap is zero, the primal optimal variabless_(min)*, s*, x* and c* can be recovered from the dual optimal variableλ*. The optimal Lagrange multiplier λ* can be obtained by updating λusing the subgradient method or the ellipsoid method. For example, inthe subgradient method, the update in the m-th step is given byλ(m+1)=[λ(m)+α_(m)(x*−c*)]⁺,  (10)

where α_(m) is the step size, and [·]+represents the projection onto thenonnegative orthant. In order to obtain the optimal flow vector x* andthe optimal capacity vector c*, the associated subproblems must besolved.

The higher-layer subproblem is precisely the first maximum in (8). It isnow necessary to characterize the feasible flow rate region

(s). Note that the network topology is fully characterized by a matrixAε{−1,0,1}K×L, whose (k, l)-element a_(kl) is given by

$\begin{matrix}{a_{kl} = \left( \begin{matrix}1 & {{{if}\mspace{14mu} l} \in {{??}(k)}} \\{- 1} & {{{if}\mspace{14mu} l} \in {{??}(k)}} \\0 & {{otherwise}.}\end{matrix} \right.} & (11)\end{matrix}$

Assume that a plain routing without network coding is employed to routethe packets. Then, given the throughput vector s,

(s) is characterized by the conditionsAx=s  (12)x

0,  (13)

where (12) represents the flow conservation constraints and (13)represents the requirement that the flow rates must be nonnegative. Theresulting higher-layer subproblem

$\begin{matrix}{{{\max\limits_{s,s_{\min},x}s_{\min}} - {\lambda^{T}x}}{{subject}\mspace{14mu}{to}\mspace{14mu} s_{\min}s}{{Ax} = s}{x \succcurlyeq 0.}} & (14)\end{matrix}$

is a linear program that can be efficiently solved.

The PHY/MAC subproblem is given by the second maximum in (8), which is aweighted capacity maximization problem with the dual prices λ_(l)playing the role of the weighting factor for link l. The capacity region

depends on the specific transmission strategy employed in the physicallayer as well as the power resource constraints imposed on the radios.In the next subsection, the capacity region without the cooperativerelaying technique is first described, which is followed by thesubsection that extends the formulation to the case with cooperativerelaying.

For simplicity of the practical receiver design, it is assumed thatjoint decoding of superposition-coded multi-user signals is notconsidered. Thus, the discussion is confined to the case of single-userdecoding where the signals not contributed by the desired user aresimply treated as interference. However, the present framework caneasily incorporate the joint decoding case.

In the cross-layer interaction of the OFDMA multi-hop network and theassociated optimization algorithms, the link capacity is modeled by aShannon-type formula that does not accurately model the specific codingand modulation schemes employed in the actual systems. Again, withrecent advances in formulating and solving the OFDMA resource allocationproblems with practical coding and modulation schemes, it is quitefeasible to incorporate these non-idealities into the present framework.

Denote the PHY link capacity of link lε

_(t) over tone n in time slot t by c_(l) ^((n,t)). The total capacityc_(l) of link l is simply given by summing the per-tone capacities overall tones and time slots: c_(l)=Σ_(n,t)c_(l) ^((n,t)). The per-tonecapacity c_(l) ^((n,t)) is modeled by the Shannon capacity formula forideal Gaussian signaling given by

$\begin{matrix}{{c_{l}^{({n,t})} \leq {\log\left( {1 + \frac{G_{ll}^{(n)}P_{l}^{({n,t})}}{{\sum\limits_{\underset{l^{\prime} \in {??}_{??}}{l^{\prime} \neq l}}{G_{{ll}^{\prime}}^{(n)}P_{l^{\prime}}^{({n,t})}}} + 1}} \right)}},{{for}\mspace{14mu}{\forall{l \in {??}_{??}}}},{\forall n},{\forall t},} & (15)\end{matrix}$

where G_(ll′) ^((n)) is the OFDM channel gain from the transmitter oflink l′ to the receiver of link l over the n-th tone normalized by thenoise variance N₀. G_(ll′) ^((n)) will be alternatively denoted in thesequel as G_(Q(l),S(l′)) ^((n)) using the node indices instead of thelink indices. P_(l) ^((n,t))≧0 is the transmit power at the transmitterof link l over tone n in time slot t.

The total transmit power at each transmitter is constrained as

$\begin{matrix}{{{\sum\limits_{\underset{l \in {??}_{??}}{l \in {{??}{(k)}}}}{\sum\limits_{n}P_{l}^{({n,t})}}} \leq {NP}_{k,\max}},{{for}\mspace{14mu}{\forall k}},{\forall n},{\forall t},} & (16)\end{matrix}$

where N is the number of the OFDM tones, P_(k,max) is the maximumaverage transmit power at node k, averaged over N OFDM tones. Thus thePHY/MAC subproblem without cooperative relaying can be written as(P1)max Σ_(n,t)Σ_(l)ε

_(t)λ_(l)c_(l) ^((n,t))subject to (15) (16),P_(l) ^((n,t))≧0,∀l,n,t.  (17)

Relaying can aid the transmission of a signal from the source to thedestination when the channel between them is weak by providing analternative path via a relay with a better channel condition.Cooperative relaying differs from conventional relaying in that a set ofdistributed relay nodes can collaborate to achieve the spatial diversityeffect of a multi-antenna array. For example, in the decode-and-forward(DF) cooperative relaying scheme, two or more relays that were able todecode the source data transmit the packet cooperatively to thedestination. In the amplify-and-forward (AF) cooperative relayingscheme, the relays simply store the received signal waveform andtransmit it simultaneously to the destination. In the setting ofmulti-hop networks, the cooperation can occur among the nodes along amulti-hop route. Employing various coding and diversity receptionstrategies, cooperative relaying can have a quite a number of differentcombinations.

To simplify the formulation, only a two-phase DF cooperative relayingscheme depicted in FIG. 4( a) is considered. First, in the broadcastphase, the source node broadcasts the data to be decoded by a cluster ofrelays. Then, in the cooperation phase, the relays cooperate intransmitting the data to the destination.

To make the cooperative relaying transparent to the higher-layersubproblem, a set of virtual links and nodes are introduced to theoriginal network topology

. Specifically, for each of the cooperating cluster of relays, a virtualnode is defined. Then, each cooperative relaying transmission consistsof two virtual links: virtual link lε

_(b) that connects the source to a virtual node and virtual link l′ε

_(c) that connects the virtual node to the destination (See FIG. 4( b),where the virtual node and links are depicted by the dashed circle andlines). Note that l and l′ correspond to the broadcast and thecooperation phases of cooperative relaying, respectively. Then, thehigher-layer subproblem can be solved in the exactly same way as for thecase without cooperative relaying, over the augmented graph

′=(

′,

′), where

′ represents the set of nodes in the augmented graph, and

′=

∪

_(b)∪

_(c) represents the set of links in the augmented graph. Note that theaugmented topology can be scheduled using the same “even-odd” framework.To stress the difference of the active link sets for differentunderlying topologies, the active link sets are denoted as a function ofthe associated graphs, e.g.,

_(t)(

′). Also, the set of cooperating relays associated with a virtual linklε

_(b)∪

_(c) is denoted by

(l).

Since the DF relaying is employed, the cooperative relays must all beable to decode the data from the source. Thus, the PHY capacity of abroadcast-phase virtual link lε

_(b) is determined by the minimum of the capacities of the individuallinks from the source to the relays:

$\begin{matrix}{{c_{l}^{({n,t})} \leq {\min\limits_{k \in {\mathcal{R}{(l)}}}c_{l,k}^{({n,t})}}},{{for}\mspace{14mu}{\forall{l \in {{{??}_{??}\left( {??}^{\prime} \right)}\bigcap\; ɛ_{b}}}}},{\forall n},{\forall t},{where}} & (18) \\{c_{l,k}^{({n,t})}\overset{\Delta}{=}{{\log\left( {1 + \frac{G_{k,{S{(l)}}}^{(n)}P_{l}^{({n,t})}}{{\sum\limits_{\underset{l^{\prime} \in {{({ɛ\bigcup\; ɛ_{b}})}\bigcap{{??}_{??}{({??}^{\prime})}}}}{l^{\prime} \neq l}}{G_{k,{S{(l^{\prime})}}}^{(n)}P_{l^{\prime}}^{({n,t})}}} + {\sum\limits_{\underset{l^{\prime} \in \;{ɛ_{c}\bigcap{{??}_{??}{({??}^{\prime})}}}}{l^{\prime} \neq l}}{\sum\limits_{k^{\prime} \in {\mathcal{R}{(l^{\prime})}}}{G_{k,k^{\prime}}^{(n)}{{\hat{P}}_{l^{\prime}}^{({n,t})}\left( k^{\prime} \right)}}}} + 1}} \right)}.}} & (19)\end{matrix}$

Here, {circumflex over (P)}_(l) ^((n,t))(k) is the transmit power of therelaying node kε

(l) associated with the cooperation-phase virtual link l. In (19), thefirst term in the denominator of the fraction inside log(·) representsthe interference from the other non-cooperative links as well as theinterference from the other cooperative links in the broadcast phase,all within the same time slot. Similarly, the second term in thedenominator represents the interference coming from the cooperativelinks in the cooperation phase.

In the cooperation phase, the capacity formula depends on the specificcooperative transmission strategy employed. If the transmit signals canbe co-phased at the relays so that the signals add up coherently at thedestination, the gain from distributed beamforming can be achieved.Under this scenario, the link capacities of the cooperation-phasevirtual links are modeled as

$\begin{matrix}{{c_{l}^{({n,t})} \leq {\log\left( {1 + \frac{\left( {\sum\limits_{k \in {\mathcal{R}{(l)}}}\sqrt{G_{{Q{(l)}},k}^{(n)}{{\hat{P}}_{l}^{({n,t})}(k)}}} \right)^{2}}{{\underset{l^{\prime} \in {{({ɛ\bigcup\; ɛ_{b}})}\bigcap{{??}_{??}{({??}^{\prime})}}}}{\sum\limits_{l^{\prime} \neq l}}{G_{{Q{(l)}},{S{(l^{\prime})}}}^{(n)}P_{l^{\prime}}^{({n,t})}}} + {\sum\limits_{\underset{l^{\prime} \in \;{ɛ_{c}\bigcap{{??}_{??}{({??}^{\prime})}}}}{l^{\prime} \neq l}}{\sum\limits_{k^{\prime} \in {\mathcal{R}{(l^{\prime})}}}{G_{{Q{(l)}},k}^{(n)}{{\hat{P}}_{l^{\prime}}^{({n,t})}\left( k^{\prime} \right)}}}} + 1}} \right)}},{{for}\mspace{14mu}{\forall{l \in {{{??}_{??}\left( {??}^{\prime} \right)}\bigcap ɛ_{c}}}}},{\forall n},{\forall{t.}}} & (20)\end{matrix}$

The non-virtual link capacity formulas must also account for theinterferences caused by the cooperative transmissions. Thus, in acooperative relaying-enabled network, (15) should be updated as

$\begin{matrix}{{c_{l}^{({n,t})} \leq {\log\left( {1 + \frac{G_{ll}^{(n)}P_{l}^{({n,t})}}{{\sum\limits_{\underset{l^{\prime} \in {{({ɛ\bigcup\; ɛ_{b}})}\bigcap{{??}_{??}{({??}^{\prime})}}}}{l^{\prime} \neq l}}{G_{{ll}^{\prime}}^{(n)}P_{l^{\prime}}^{({n,t})}}} + {\underset{l^{\prime} \in \;{ɛ_{c}\bigcap{{??}_{??}{({??}^{\prime})}}}}{\sum\limits_{l^{\prime} \neq l}}{\sum\limits_{k^{\prime} \in {\mathcal{R}{(l^{\prime})}}}{G_{{Q{(l)}},k^{\prime}}^{(n)}{{\hat{P}}_{l^{\prime}}^{({n,t})}\left( k^{\prime} \right)}}}} + 1}} \right)}},{{for}\mspace{14mu}{\forall{l \in {{??}_{??}({??})}}}},{\forall n},{\forall{t.}}} & (21)\end{matrix}$

Also, the power constraints at each node must reflect the cooperativetransmissions. This can be modeled by

$\begin{matrix}{{{\sum\limits_{n}\left( {{\sum\limits_{\underset{l \in {{({ɛ\bigcup\; ɛ_{b}})}\bigcap{{??}_{??}{({??}^{\prime})}}}}{l \in {{??}{(k)}}}}P_{l}^{({n,t})}} + {\sum\limits_{\underset{k \in {\mathcal{R}{(l)}}}{l \in \;{ɛ_{c}\bigcap{{??}_{??}{({??}^{\prime})}}}}}{{\hat{P}}_{l}^{({n,t})}(k)}}} \right)} \leq {NP}_{k,\max}},{{for}\mspace{14mu}{\forall k}},{\forall{t.}}} & (22)\end{matrix}$

The PHY subproblem with cooperative relaying transmissions can bewritten as(P2)max Σ_(n,t)Σ_(lε)

(

) λ_(l)c_(l) ^((n,t))subject to (18)-(22)P _(l) ^((n,t))≧0,∀lε

_(t)(

′)∩(

∪ε_(b)),∀n,∀t{circumflex over (P)} _(l) ^((n,t))(k)≧0,∀lε

(

) ∩ε _(c) ,∀kε

(l),∀n,∀t.  (23)(24)

The PHY/MAC subproblems (P1) and (P2) must be solved for the optimalpower profile over the entire OFDMA band for each link and each timeslot. As the number of OFDMA tones and links in the network grows large,it becomes essential to seek ways to reduce the computationalcomplexity. First, it can be observed that the problems are actuallydecoupled in different time slots in our formulation. Therefore, theproblems can be solved independently for each time slot t=0 and t=1.

Secondly, it is observed that the total power constraints are the onlyconstraints that couple the problems over different OFDM tones.Therefore, by relaxing the total power constraints by the Lagrangemultiplier technique, one can also decouple the problem into N per-toneproblems that can be solved independently given the Lagrangemultipliers. Then, the computational complexity becomes linear in N,which is a significant reduction especially when N is large. However,the PHY/MAC subproblems (P1) and (P2) are not convex in general due tothe non-convexity of the link capacity formulas when interference ispresent. Therefore, strong duality may not hold and the dual approachmay not yield the optimal solution.

Recently, it was discovered that in multicarrier communication systems,the duality gap of the (potentially non-convex) weighted sum capacitymaximization problems under a total power constraint approaches zerowhen the number of the tones N goes to infinity. The intuition is thatas N becomes large compared to the number of multipaths in the channel,there are many tones that share similar channel gains with theneighboring tones. These group of tones can approximate the effect of“time-sharing” in the frequency domain to “convexity” the problem. Dueto this result, one can still use the dual method to solve thesubproblems (P1) and (P2) for complexity reduction, and, in fact, canclaim that it actually performs closer to the optimum as N gets large.

It is noteworthy that strong duality holds regardless of the convexitystructure of the original problem. Therefore, the problem canincorporate additional non-convex constraints such as finite number ofavailable power and rate levels, or allowing at most one link per tonefor pure OFDMA channel allocation.

With the total power constraints (16) relaxed by introducing a set ofdual variables {μ_(k) ^((t))}, the Lagrangian for (P1) can be written as

$\begin{matrix}{{\mathcal{L}_{P\; 1}\left( {\left\{ c_{l}^{({n,t})} \right\},\left\{ P_{l}^{({n,t})} \right\},\left\{ \mu_{k}^{(t)} \right\}} \right)} = {{\sum\limits_{n,t}{\sum\limits_{l \in {??}_{??}}{\lambda_{l}c_{l}^{({n,t})}}}} - {\sum\limits_{t,k}{{\mu_{k}^{(t)}\left( {{\sum\limits_{n}{\sum\limits_{\underset{l \in {??}_{??}}{l \in {{??}{(k)}}}}P_{l}^{({n,t})}}} - {NP}_{k,\max}} \right)}.}}}} & (25)\end{matrix}$

The dual objective is given by

ℊ 1 ⁡ ( { μ k ( t ) } ) = ∑ n , t ⁢ max ⁢ { ∑ ℓ ∈ t ⁢ λ ℓ ⁢ c ℓ ( n , t ) - ∑k ⁢ μ k ( t ) ⁢ ∑ ℓ ∈ ?? ⁡ ( k ) ℓ ∈ t ⁢ P ℓ ( n , t ) | , } + ∑ t , k ⁢ μ k( t ) ⁢ NP k , max , ( 26 )

where it is evident that the problem can be solved for each tone andtime slot.

In summary, the decomposed problem for (P1) for tone n and time slot tis given by

max ⁢ ∑ ℓ ∈ ⁢ λ ℓ ⁢ ℓ ( n , t ) - ∑ k ⁢ μ k ( t ) ⁢ ∑ ℓ ∈ ⁢ ( k ) ℓ ∈ t ⁢ P ℓ (n , t ) ⁢ ⁢ subject ⁢ ⁢ to ⁢ ⁢ ( 15 ) , ( 17 ) . ( 27 )

Similarly, the decomposed problem for (P2) for tone n and time slot t isgiven by

max ⁢ ∑ ℓ ∈ t ⁢ ( ′ ) ⁢ λ ℓ ⁢ c ℓ ( n , t ) - ∑ k ⁢ μ k ( t ) ( ∑ ℓ ∈ ⁢ ( k )ℓ ∈ ( ⋃ b ) ⋂ t ⁢ ( ′ ) ⁢ P ℓ ( n , t ) + ∑ ℓ ∈ c ⋂ t ⁢ ( ′ ) k ∈ ⁢ ( ℓ ) ⁢ P^ l ( n , t ) ⁡ ( k ) ) ( 28 )

Subject to (18)-(21) and (23)-(24)

Although a hierarchical decomposition approach of first decomposing theproblem (1) into two subproblems in different layers and then furtherdecomposing the PHY/MAC subproblems (P1) and (P2) into the per-toneproblems, this is merely for convenience of exposition. In actualimplementation of the optimization algorithm, the decomposition can bedone without such a hierarchy so that all the subproblems can be solvedin parallel and the dual prices λ and μ can be updated simultaneouslyfor faster convergence.

Next, the solution methods for the per-tone PHY/MAC subproblems will beaddressed including various practical issues in implementing theoptimization algorithms. Also, closed-form solutions for the per-toneproblems are obtained in some special cases.

The optimal primal variables can be recovered in principle by maximizingthe Lagrangian given the dual prices, i.e., by solving the associatedsubproblems. However, this must be done with care since there can bemultiple maximizers of the Lagrangian, and some of them might not befeasible solutions to the original problem.

This is manifested in our problem setup most clearly in solving thePHY/MAC subproblems for those links whose transmitters do not meet thepower constraints with equality. Note that in multi-hop networks,bottleneck links often appear (usually near the gateway) because all thenetwork traffic is eventually siphoned off at the gateway. Due to thebottleneck effect, the remaining parts of the network often have unusedresources. For those nodes k whose power constraints are slack in timeslot t, the optimal dual price λ_(k) ^((t))* must be zero from thecomplementary slackness condition. Then, it can be verified that theoptimal dual prices λ_(l)* for lε

(k)∩

_(t) could also be zero. For example, if a link lε

(k)∩

_(t) does not interfere with any other link in the time slot, λ_(l) mustbe zero. When this is the case, the optimal primal solution cannot berecovered from the optimal dual prices.

To prevent this, a small term −εΣ_(l,n,t)P_(l) ^((n,t)) is added to theobjective of (P1), where ε>0 is a small constant. Similarly, −εΣ_(lε)

_(∪)

_(b) _(,n,t) P_(l) ^((n,t))−εΣ_(lε)

_(c) _(,n,t)Σ_(kε)

_((l)) {circumflex over (P)}_(l) ^((n,t))(k) is added to the objectiveof (P2). Then, the per-tone problems that are solved for (P1) and (P2)are given by replacing μ_(k) ^((t)) with (μ_(k) ^((t))+

) in (27) and (28), respectively. This replacement has the followinginterpretation: whenever the nodes have slack power constraints and thecorresponding dual prices become very small, the optimization isautomatically switched to the total transmit power minimization for theslack nodes.

The per-tone problem (27) was considered in numerous references albeitthe context may be different. A number of different numerical techniquesto solve it were proposed in the literature. In, a discretizedexhaustive search was used, which has a merit in that a solution quiteclose to the global optimum can be obtained if the discretization isfine enough. However, the complexity is exponential in the number of thelinks in the network, which makes it feasible only for a very smallnetwork. A coordinate descent search was used in to reduce thecomplexity. The search converges to a local optimum. A game theoreticapproach was considered, where each player (link) maximizes a payofffunction that approximates her interference contribution to the otherplayers to the first order. The game can be shown to converge to a localoptimum of the original problem. When the SINRs in the capacity formulasare much larger than 0 dB, e.g., by a large processing gain in thespread-spectrum systems, a high SINR approximation can be employed totransform the original problem into a geometric program (GP), which inturn can be transformed to a convex optimization problem. In the mid- orlow SINR regime, iterative procedures were developed that solve a seriesof convex optimization problems. The algorithms can also be shown toconverge to a local optimum of the original problem.

Although these techniques provide quite efficient means to find a localoptimum of the per-tone subproblems, they often caused a glitch in theoverall dual method in our numerical experiments. For example, when thedual update was done by the ellipsoid method, the optimization algorithmoften converged to an infeasible solution even with a sufficiently largeN. In fact, only the discretized exhaustive search yielded satisfactorysolutions for all the test problems that we experimented. Obviously,this is caused by non-idealities of our problem formulation, whichassumed a zero duality gap for the PHY/MAC subproblems.

The convergence behavior of the dual method for a non-convex NUM problemwas studied in. An important theorem proved in the work states that ifthe primal optimal variables x*(λ) recovered by optimizing theLagrangian

(x,λ) is continuous with respect to λ at the optimal dual price λ=λ*,then strong duality holds and the dual method converges to the globaloptimum.

Based on this result, an intuitive explanation as to why the algorithmsthat yield a local optimum of the per-tone problem often cause anon-zero duality gap is as follows. In order for the recovered primaloptimal variables {P_(l) ^((n,t))*} to be (almost) continuous withrespect to the dual prices {μ_(k) ^((t))}, a small change in the valuesof {μ_(k) ^((t))} should not cause a large portion of the N per-toneproblems to switch to a different local optimum abruptly. However, ifall the per-tone problems are exploring only the vicinity of a singlestarting point, the chances of such cases will be high.

In order to avoid the exponential complexity of the discretizedexhaustive search yet still to obtain a feasible solution, a couple ofheuristic remedies are adopted. First, by noting that the set of highlyinterfering links will not be active at the same time at the optimalsolution, the link activation sets

_(t) are partitioned into a small number of subsets with at most severallinks in each subset, and each of these subsets are consideredseparately. For example, the links that share the same transmitters orreceivers will most likely not share the same OFDM tone due to excessiveinterference. The per-tone problems can be solved for each of thesesubsets and the one with the largest optimum value is selected. Then, asimple gradient projection method can be used with multiple randomstarting points. The gradient projection method still yields a localoptimum, but due to the multiple starting points, the per-tone problemsolver can examine more diversified search space. In our numericalresults, even with a small number of starting points (say, N_(S)=5), thedual method yielded feasible solutions.

In the case of using cooperative relaying, the gradient search methodcannot be applied to solve (28) directly. The reason is that since thecapacity formula (18) for the broadcast-phase operation involves aminimum of functions, it is not differentiable. A straightforward way tosolve a non-differentiable optimization problem is to use thesubgradient method. However, the subgradient method is usually quiteslow.

In this work, a smooth approximation of the minimum can approximate

$\begin{matrix}{{\min\limits_{k \in {(\ell)}}{c_{\ell,k}^{({n,t})}(P)}}{by}} & (29) \\{{{- \frac{1}{d}}\log\left\{ {\sum\limits_{k \in {(l)}}{w_{k}{\mathbb{e}}^{- {{dc}_{\ell,k}^{({n,t})}{(P)}}}}} \right\}},} & (30)\end{matrix}$where P is the vector of the relevant optimization variablesrepresenting the transmit power, d is a positive constant and {w_(k)}satisfy Σ_(kεR(l))w_(k)=1 and w_(k)≧0. The approximation can be madeincreasingly accurate by updating the parameter d and the weights{w_(k)} according to the expressions

$\begin{matrix}{{d\left( {m^{\prime} + 1} \right)} = {\beta\;{d\left( m^{\prime} \right)}}} & (31) \\{{{w_{k}\left( {m^{\prime} + 1} \right)} = \frac{{w_{k}\left( m^{\prime} \right)}{\mathbb{e}}^{{- {d{(m^{\prime})}}}{c_{\ell,k}^{({n,t})}{({P^{*}{(m^{\prime})}})}}}}{\sum\limits_{k^{\prime} \in {(\ell)}}{{w_{k^{\prime}}\left( m^{\prime} \right)}{\mathbb{e}}^{{- {d{(m^{\prime})}}}{c_{\ell,k^{\prime}}^{({n,t})}{({P^{*}{(m^{\prime})}})}}}}}},} & (32)\end{matrix}$

where β>1 is a constant, m′ is the iteration count, and P*(m′) is theoptimal power vector obtained from solving the m′-th approximateproblem. Thus, to solve the per-tone problem (28), a series ofapproximated problems are solved until convergence.

Next, solutions for pure OFDMA are discussed. When no spatial reuse ofthe OFDM tones is allowed, i.e., when pure OFDMA is enforced,closed-form solutions can be obtained for the per-tone problems. Sinceonly one link can be activated on a particular tone in a particular timeslot, essentially the problem is to find the link for each tone and timeslot that maximizes the objective of the per-tone problems.

If a non-virtual link lε

_(t) is activated on tone n in time slot t, the optimal power can bereadily obtained from the KKT conditions, and are given by

$\begin{matrix}{P_{\ell}^{{({n,t})}*} = {\left\lbrack {\frac{\lambda_{\ell}}{\mu_{S{(\ell)}}^{(t)} + ɛ} - \frac{1}{G_{\ell\ell}^{(n)}}} \right\rbrack^{+}.}} & (33)\end{matrix}$

When a cooperative relaying link is activated, the optimal powerallocation is determined in a simple closed form provided that thenumber of relays is two (|

(l)|=2). If a broadcast-phase virtual link lε

_(b) is activated on tone n in time slot t, the optimal transmit poweris calculated to be

$\begin{matrix}{{P_{\ell}^{{({n,t})}*} = \left\lbrack {\frac{\lambda_{\ell}}{\mu_{S{(\ell)}}^{(t)} +} - \frac{1}{G_{k^{*},{S{(\ell)}}}^{(n)}}} \right\rbrack^{+}},} & (34)\end{matrix}$

where k* is the node associated with the minimum capacity link, i.e.,

$\begin{matrix}{{k^{*} = {\arg\mspace{11mu}{\min\limits_{k \in {(\ell)}}{c_{\ell,k}^{({n,t})}.}}}}\;} & (35)\end{matrix}$

For a cooperation-phase virtual link lεE_(c), the optimal powers at thetwo cooperating relays k_(i)εR(l), i=1, 2 are given by

$\begin{matrix}{{{{\hat{P}}_{\ell}^{{({n,t})}*}\left( k_{i} \right)} = {\frac{1}{\eta_{i}^{2}}\left\lbrack {\frac{\lambda_{\ell}\eta_{i}}{\mu_{k_{i}}^{(t)} +} - \frac{1}{G_{{Q{(\ell)}},k_{i}}^{(n)}}} \right\rbrack}^{+}},{where}} & (36) \\{\eta_{i} = {1 + {\frac{\mu_{k_{i}}^{(t)} +}{\mu_{k_{3 - i}}^{(t)} +}{\frac{G_{{Q{(\ell)}},k_{3 - i}}^{(n)}}{G_{{Q{(\ell)}},k_{i}}^{(n)}}.}}}} & (37)\end{matrix}$

Then, the index of the link l*ε

_(t)(

′) that is chosen to be activated is the maximizer of the objective:

ℓ * = arg ⁢ max ℓ ⁢ ⁢ { λ ℓ ⁢ ⁢ log ⁡ ( 1 + G ℓℓ ( n ) ⁢ P ℓ ( n , t ) * ) - μS ⁡ ( ℓ ) ( t ) ⁢ P ℓ ( n , t ) * ⁢ | ℓ ∈ t ⁢ ( ) , ⁢ ⁢ λ ℓ ⁢ ⁢ log ⁡ ( 1 + G k *, S ⁡ ( ℓ ) ( n ) ⁢ P ℓ ( n , t ) * ) - μ S ⁡ ( ℓ ) ( t ) ⁢ P ℓ ( n , t ) * ⁢| ℓ ∈ t ⁢ ( ′ ) ⋂ b , ⁢ λ ℓ ⁢ ⁢ log ( 1 + ( ∑ k ∈ ⁢ ( ℓ ) ⁢ G Q ⁡ ( ℓ ) , k ( n) ⁢ P ^ ℓ ( n , t ) * ⁡ ( k ) ) 2 ) - ∑ k ∈ ⁢ ( ℓ ) ⁢ μ k ( t ) ⁢ P ^ ℓ ( n ,t ) * ⁡ ( k ) ⁢ | ℓ ∈ t ⁢ ( ′ ) ⋂ c } . ( 38 )

The optimal resource allocation problem for OFDMA wireless multi-hopnetworks was studied by modeling the cross-layer interaction of routing,MAC and physical layer. In addition to the complexity of the multi-layeroptimization, the OFDMA air interface incurs significant challenge tothe problem due to the large number of subchannels to which theresources must be optimally distributed. Moreover, when mutualinterference between the links is modeled for maximal spatial reuse ofthe spectral resources, the problem becomes non-convex and is difficultto solve. The dual decomposition technique decouples the problem intodifferent layers and OFDM tones without losing optimality when thenumber of the tones is large. The resulting subproblems can beseparately solved under the coordination of the dual prices that linkthe subproblems, which reduces the computational complexitysignificantly. By exploiting the flexibility of this framework, thecooperative relaying technique improves the balanced end-to-endthroughput of the network. Under the practical half-duplex constraintsof the radios, the cooperative relaying identifies a unique avenue toalleviate the bottleneck phenomenon that typically manifests itself inmulti-hop networks. That is, the cooperative relaying can exploit theunused resources in the non-bottleneck links to aid the links that arein bottleneck condition, which gives a net increase in the overallthroughput.

1. A method to allocate one or more resources in a multi-hop network,comprising: a. solving one or more higher-layer sub-problems; b. solvingone or more physical layer and media access control (PHY/MAC) layersub-problems per tone per time slot with one of cooperative relaying ofradio signals; or spatial reusing of radio spectrum; and no cooperativerelaying of radio signals and no spatial reusing of the radio spectrum;c. updating prices; and d. allocating radio resources based on thePHY/MAC layer sub-problems;$P_{\ell}^{{({n,t})}*} = {\left\lbrack {\frac{\lambda_{\ell}}{\mu_{S{(\ell)}}^{(t)} + ò} - \frac{1}{G_{\ell\ell}^{(n)}}} \right\rbrack^{+}\mspace{14mu}{including}\mspace{20mu}{solving}}$ℓ * = arg ⁢ ⁢ max ℓ ⁢ { λ ℓ ⁢ ⁢ log ⁡ ( 1 + G ℓℓ ( n ) ⁢ P ℓ ( n , t ) * ) - μS ⁡ ( ℓ ) ( t ) ⁢ P ℓ ( n , t ) * ⁢ | ℓ ∈ t ⁢ ( ) } .
 2. The method of claim1, comprising solving one or more physical layer and media accesscontrol (PHY/MAC) layer sub-problems per tone per time slot withcooperative relaying of radio signals and no spatial reusing of radiospectrum.
 3. The method of claim 2, comprising:$P_{\ell}^{{({n,t})}^{*}} = {{\left\lbrack {\frac{\lambda_{\ell}}{\mu_{S{(\ell)}}^{(t)} + ò} - \frac{1}{G_{k^{*},{S{(\ell)}}}^{(n)}}} \right\rbrack^{+}{{\hat{P}}_{\ell}^{{({n,t})}^{*}}\left( k_{i} \right)}} = {\frac{1}{\eta_{i}^{2}}\left\lbrack {\frac{\lambda_{\ell}\eta_{i}}{\mu_{k_{i}}^{(t)} + ò} - \frac{1}{G_{{Q{(\ell)}},k_{i}}^{(n)}}} \right\rbrack}^{+}}$ℓ * = arg ⁢ ⁢ max ℓ ⁢ { λ ℓ ⁢ ⁢ log ⁡ ( 1 + G ℓℓ ( n ) ⁢ P ℓ ( n , t ) * ) - μS ⁡ ( ℓ ) ( t ) ⁢ P ℓ ( n , t ) * ⁢ | ℓ ∈ , ⁢ λ ℓ ⁢ ⁢ log ⁡ ( 1 + G k * , S ⁡ (ℓ ) ( n ) ⁢ P ℓ ( n , t ) * ) - μ S ⁡ ( ℓ ) ( t ) ⁢ P ℓ ( n , t ) * ⁢ | ℓ ∈t ⁡ ( ′ ) ⋂ b , ⁢ λ ℓ ⁢ ⁢ log ( 1 + ( ∑ k ∈ ⁢ ( ℓ ) ⁢ G Q ⁡ ( ℓ ) , k ( n ) ⁢ P^ ℓ ( n , t ) * ⁡ ( k ) ) 2 ) - ∑ k ∈ ⁢ ( ℓ ) ⁢ μ k ( t ) ⁢ P ^ ℓ ( n , t) * ⁡ ( k ) ⁢ | ℓ ∈ } .
 4. The method of claim 1, comprising solving oneor more physical layer and media access control (PHY/MAC) layersub-problems per tone per time slot with no cooperative relaying ofradio signals but with spatial reusing of radio spectrum.
 5. The methodof claim 4, comprising solving max ⁢ ∑ ℓ ∈ ⁢ t ⁢ λ ℓ ⁢ c ℓ ( n , t ) - ∑ k ⁢( μ ℓ ( t ) + ò ) ⁢ ∑ ℓ ∈ ⁢ ( k ) ℓ ∈ t ⁢ P ℓ ( n , t ) subject ⁢ ⁢ to ⁢ ⁢ P ℓ( n , t ) ≥ 0 , c ℓ ( n , t ) ≤ log ( 1 + G ℓℓ ( n ) ⁢ P ℓ ( n , t ) ∑ ℓ′ ≠ ℓ ℓ ′ ∈ t ⁢ G ℓℓ ′ ( n ) ⁢ P ℓ ′ ( n , t ) + 1 ) numerically withmultiple initial points.
 6. The method of claim 1, comprising solvingone or more physical layer and media access control (PHY/MAC) layersub-problems per tone per time slot with cooperative relaying of radiosignals and spatial reusing of radio spectrum.
 7. The method of claim 6,comprising solving max ⁢ ∑ ℓ ∈ t ⁡ ( ′ ) ⁢ λ ℓ ⁢ c ℓ ( n , t ) - ∑ k ⁢ ( μ ℓ( t ) + ò ) ⁢ ( ∑ ℓ ∈ ⁢ ( k ) ℓ ∈ ( ⋃ b ) ⋂ t ⁡ ( ′ ) ⁢ P ℓ ( n , t ) + ∑ ℓ∈ k ∈ ⁢ ( ℓ ) ⁢ P ^ ℓ ( n , t ) ⁡ ( k ) ) ⁢ s . t . ⁢ c ℓ ( n , t ) ≤ min k ∈⁢( ℓ ) ⁢ c ℓ , k ( n , t ) , ∀ ℓ ∈ t ⁡ ( ′ ) ⋂ b , ⁢ c ℓ , k ( n , t ) ⁢ = Δ ⁢log ( 1 + G k , S ⁡ ( ℓ ) ( n ) ⁢ P ℓ ( n , t ) ∑ ℓ ′ ≠ ℓ ℓ ′ ∈ ( ⋃ b ) ⋂t ⁡ ( ′ ) ⁢ G k , S ⁡ ( ℓ ′ ) ( n ) ⁢ P ℓ ′ ( n , t ) + ∑ ℓ ′ ≠ ℓ ℓ ′ ∈ c ⋂t ⁡ ( ′ ) ⁢ ∑ k ′ ∈ ⁢ G k , k ′ ( n ) ⁢ P ^ ℓ ′ ( n , t ) ⁡ ( k ′ ) + 1 ) c ℓ( n , t ) ≤ log ( 1 + ( ∑ k ∈ ⁢ ( ℓ ) ⁢ G Q ⁡ ( ℓ ) , k ( n ) ⁢ P ^ ℓ ( n ,t ) ⁡ ( k ) ) 2 ∑ ℓ ′ ≠ ℓ ℓ ′ ∈ ( ⋃ b ) ⋂ t ⁡ ( ′ ) ⁢ G Q ⁡ ( ℓ ) , S ⁡ ( ℓ ′) ( n ) ⁢ P ℓ ′ ( n , t ) + ∑ ℓ ′ ≠ ℓ ℓ ′ ∈ c ⋂ t ⁡ ( ′ ) ⁢ ∑ k ′ ∈ ⁢ ( ℓ ′) ⁢ G Q ⁡ ( ℓ ) , k ′ ( n ) ⁢ P ^ ℓ ′ ( n , t ) ⁡ ( k ′ ) + 1 ) , ⁢ ∀ ℓ ∈ t ⁡( ′ ) ⋂ c c ℓ ( n , t ) ≤ log ( 1 + G ℓℓ ( n ) ⁢ P ℓ ( n , t ) ∑ ℓ ′ ≠ ℓℓ ′ ∈ ( ⋃ b ) ⋂ t ⁡ ( ′ ) ⁢ G ℓℓ ′ ( n ) ⁢ P ℓ ′ ( n , t ) + ∑ ℓ ′ ≠ ℓ ℓ ′∈ c ⋂ t ⁡ ( ′ ) ⁢ ∑ k ′ ∈ ⁡ ( ℓ ′ ) ⁢ G Q ⁡ ( ℓ ) , k ′ ( n ) ⁢ P ^ ℓ ′ ( n ,t ) ⁡ ( k ′ ) + 1 ) , ∀ ℓ ∈ t ⁡ ( ′ ) numerically with multiple initialpoints.
 8. The method of claim 7, comprising solving an approximateproblem given by max ⁢ ∑ ℓ ∈ t ⁢ ( ′ ) ⁢ λ ℓ ⁢ c ℓ ( n , t ) - ∑ k ⁢ ( μ ℓ (t ) + ò ) ⁢ ( ∑ ℓ ∈ ⁢ ( k ) ℓ ∈ ( ⋃ b ) ⋂ ⁢ P ℓ ( n , t ) + ∑ ℓ ∈ c ⋂ t ⁢ (′ ) k ∈ ⁢ ( ℓ ) ⁢ P ^ ℓ ( n , t ) ⁡ ( k ) ) ⁢ s . t . ⁢ c ℓ ( n , t ) ≤ - 1 d⁢log ⁡ ( ∑ k ∈ ⁢ ( ℓ ) ⁢ w k ⁢ ⅇ - dc ℓ , k ( n , t ) ⁡ ( P ) ) , ∀ ℓ ∈ t ⁢ ( ′) ⋂ b , ⁢ c ℓ , k ( n , t ) ⁢ = Δ ⁢ log ( 1 + G k , S ⁡ ( ℓ ) ( n ) ⁢ P ℓ ( n, t ) ∑ ℓ ′ ≠ ℓ ℓ ′ ∈ ( ⋃ b ) ⋂ t ⁡ ( ′ ) ⁢ G k , S ⁡ ( ℓ ′ ) ( n ) ⁢ P ℓ ′( n , t ) + ∑ ℓ ′ ≠ ℓ ℓ ′ ∈ c ⋂ t ⁡ ( ′ ) ⁢ ∑ k ′ ∈ ⁢ ( ℓ ′ ) ⁢ G k , k ′ (n ) ⁢ P ^ ℓ ′ ( n , t ) ⁡ ( k ′ ) + 1 ) c ℓ ( n , t ) ≤ log ( 1 + ( ∑ k ∈ ⁢( ℓ ) ⁢ G Q ⁡ ( ℓ ) , k ( n ) ⁢ P ^ ℓ ( n , t ) ⁡ ( k ) ) 2 ∑ ℓ ′ ≠ ℓ ℓ ′ ∈( ⋃ b ) ⋂ t ⁡ ( ′ ) ⁢ G Q ⁡ ( ℓ ) , S ⁡ ( ℓ ′ ) ( n ) ⁢ P ℓ ′ ( n , t ) + ∑ ℓ′ ≠ ℓ ℓ ′ ∈ c ⋂ t ⁡ ( ′ ) ⁢ ∑ k ′ ∈ ⁢ ( ℓ ′ ) ⁢ G Q ⁡ ( ℓ ) , k ′ ( n ) ⁢ P ^ℓ ′ ( n , t ) ⁡ ( k ′ ) + 1 ) , ⁢ ∀ ℓ ∈ t ⁡ ( ′ ) ⋂ c ℓ ( n , t ) ≤ log (1 + G ℓℓ ( n ) ⁢ P ℓ ( n , t ) ∑ ℓ ′ ≠ ℓ ℓ ′ ∈ ( ⋃ b ) ⋂ t ⁡ ( ′ ) ⁢ G ℓℓ ′( n ) ⁢ P ℓ ′ ( n , t ) + ∑ ℓ ′ ≠ ℓ ℓ ′ ∈ c ⋂ t ⁡ ( ′ ) ⁢ ∑ k ′ ∈ ⁢ ( ℓ ′ ) ⁢G Q ⁡ ( ℓ ) , k ′ ( n ) ⁢ P ^ ℓ ′ ( n , t ) ⁡ ( k ′ ) + 1 ) , ∀ ℓ ∈ t ⁡ ( ′) .
 9. The method of claim 1, comprising repeating (a)-(c) untilconvergence.
 10. The method of claim 1, wherein the solving thehigher-layer sub-problem comprises solving:maxs_(min)−λ^(T)xsubject to s_(min)≦sAx=sx


0. 11. A system to allocate one or more resources in a multi-hopnetwork, comprising: one or more base stations; and a radio resourcecontroller coupled to the one or more base stations, the radio resourcecontroller including computer readable code to: solve one or morehigher-layer sub-problems; solve one or more physical layer and mediaaccess control (PHY/MAC) layer sub-problems per tone per time slot withone of: cooperative relaying of radio signals; or spatial reusing ofradio spectrum; and no cooperative relaying of radio signals and nospatial reusing of the radio spectrum; update prices; and allocate radioresources based on the PHY/MAC layer sub-problems;${P_{l}^{{({n,t})}*} = \left\lbrack {\frac{\lambda_{l}}{\mu_{S{(l)}}^{(t)} + ò} - \frac{1}{G_{ll}^{(n)}}} \right\rbrack^{+}}\mspace{14mu}$including  solving$l^{*} = {\arg\mspace{14mu}{\max\limits_{l}{\left\{ {{{\lambda_{l}{\log\left( {1 + {G_{ll}^{(n)}P_{l}^{{({n,t})}*}}} \right)}} - {\mu_{S{(l)}}^{(t)}P_{l}^{{({n,t})}*}}}❘_{l \in {A_{t}{(G)}}}} \right\}.}}}$12. The system of claim 11, comprising code to solve one or morephysical layer and media access control (PHY/MAC) layer sub-problems pertone per time slot with no cooperative relaying of radio signals and nospatial reusing of the radio spectrum.
 13. The system of claim 11,comprising code to solve one or more physical layer and media accesscontrol (PHY/MAC) layer sub-problems per tone per time slot withcooperative relaying of radio signals and no spatial reusing of radiospectrum.
 14. The system of claim 11, comprising code to solve one ormore physical layer and media access control (PHY/MAC) layersub-problems per tone per time slot with no cooperative relaying ofradio signals but with spatial reusing of radio spectrum.
 15. The systemof claim 11, comprising code to solve one or more physical layer andmedia access control (PHY/MAC) layer sub-problems per tone per time slotwith cooperative relaying of radio signals and spatial reusing of radiospectrum.
 16. A method to allocate one or more resources in a multi-hopnetwork, comprising: a. solving one or more higher-layer sub-problem(s);b. solving one or more physical layer and media access control (PHY/MAC)layer sub-problems per tone per time slot with one of cooperativerelaying of radio signals; spatial reusing of radio spectrum; and nocooperative relaying of radio signals but with spatial reusing of theradio spectrum; c. updating prices; and d. allocating radio resourcesbased on the PHY/MAC layer sub-problems;${\max{\sum\limits_{l \in A_{t}}^{\;}\;{\lambda_{l}c_{l}^{({n,t})}}}} - {\sum\limits_{k}^{\;}\;{\left( {\mu_{l}^{(t)} + ò} \right){\sum\limits_{\underset{l \in A_{t}}{l \in {O{(k)}}}}^{\;}\; P_{l}^{({n,t})}}}}$including  solving${{{subject}\mspace{14mu}{to}\mspace{14mu} P_{l}^{({n,t})}} \geq 0},{c_{l}^{({n,t})} \leq {{\log\left( {1 + \frac{G_{ll}^{(n)}P_{l}^{({n,t})}}{{\sum\limits_{\underset{l^{\prime} \in A_{t}}{l^{\prime} \neq l}}^{\;}\;{G_{{ll}^{\prime}}^{(n)}P_{l^{\prime}}^{({n,t})}}} + 1}} \right)}.}}$